Show that $8$ is an eigen value of $A$

Question

Consider the following matrix:

$$A=\begin{bmatrix}B&&&C\\D&&&F\end{bmatrix}$$ where

$$B= \begin{bmatrix} 9 &1&1&1&1\\1&9&1&1&1\\1&1&9&1&1\\1&1&1&9&1\\1&1&1&1&9\end{bmatrix}$$ and $$C=\begin{bmatrix}1&1&1&1&1\\1&1&1&1&1\\1&1&1&1&1\\1&1&1&1&1\\1&1&1&1&1\end{bmatrix}$$

and $$D=C^T$$

and $$F=\begin{bmatrix} G&H \\I &J\end{bmatrix}$$

where $$G=\begin{bmatrix} 8 &1&1&1\\1&8&1&1\\1&1&8&1\\1&1&1&8\end{bmatrix}$$ and $$H=\begin{bmatrix} 0\\0\\0\\0\end{bmatrix}$$ and $$I=H^T$$ and

$$J=\begin{bmatrix} 5\end{bmatrix}$$

Show that $8$ is an eigen value of the matrix $A$.

How should I try to prove it?Please give some hints .I dont want a complete solution

Answer 1

Hint:

• Examine the first two rows of $A-8I$, they are identical, what can you conclude.

Same result holds if you replace $D$ and $F$ with other matrices of the same size.